Introduction to Stochastic Calculus

1. Front Matter

   Pages i-xiii

   PDF

2. Discrete Parameter Martingales

   • Rajeeva L. Karandikar, B. V. Rao

   Pages 1-33

3. Continuous-Time Processes

   • Rajeeva L. Karandikar, B. V. Rao

   Pages 35-63

4. The Ito’s Integral

   • Rajeeva L. Karandikar, B. V. Rao

   Pages 65-87

5. Stochastic Integration

   • Rajeeva L. Karandikar, B. V. Rao

   Pages 89-160

6. Semimartingales

   • Rajeeva L. Karandikar, B. V. Rao

   Pages 161-213

7. Pathwise Formula for the Stochastic Integral

   • Rajeeva L. Karandikar, B. V. Rao

   Pages 215-220

8. Continuous Semimartingales

   • Rajeeva L. Karandikar, B. V. Rao

   Pages 221-249

9. Predictable Increasing Processes

   • Rajeeva L. Karandikar, B. V. Rao

   Pages 251-302

10. The Davis Inequality

    • Rajeeva L. Karandikar, B. V. Rao

    Pages 303-320

11. Integral Representation of Martingales

    • Rajeeva L. Karandikar, B. V. Rao

    Pages 321-360

12. Dominating Process of a Semimartingale

    • Rajeeva L. Karandikar, B. V. Rao

    Pages 361-381

13. SDE Driven by r.c.l.l. Semimartingales

    • Rajeeva L. Karandikar, B. V. Rao

    Pages 383-410

14. Girsanov Theorem

    • Rajeeva L. Karandikar, B. V. Rao

    Pages 411-434

15. Back Matter

    Pages 435-441

    PDF

This book sheds new light on stochastic calculus, the branch of mathematics that is most widely applied in financial engineering and mathematical finance. The first book to introduce pathwise formulae for the stochastic integral, it provides a simple but rigorous treatment of the subject, including a range of advanced topics. The book discusses in-depth topics such as quadratic variation, Ito formula, and Emery topology. The authors briefly addresses continuous semi-martingales to obtain growth estimates and study solution of a stochastic differential equation (SDE) by using the technique of random time change. Later, by using Metivier–Pellaumail inequality, the solutions to SDEs driven by general semi-martingales are discussed. The connection of the theory with mathematical finance is briefly discussed and the book has extensive treatment on the representation of martingales as stochastic integrals and a second fundamental theorem of asset pricing. Intended for undergraduate- and beginning graduate-level students in the engineering and mathematics disciplines, the book is also an excellent reference resource for applied mathematicians and statisticians looking for a review of the topic.

• Stochastic Calculus
• Martingale Convergence Theorem
• Continuous Time Process
• The Ito Integral
• Stochastic Integration
• Semimartingales

“The style is compact and clear. The presentation is well complemented by a large number of useful remarks and exercises. Graduate students attending university courses in modern probability theory and its applications can benefit a lot from working with this book. There are good reasons to expect that the book will be met positively by students, university teachers and young researchers.” (Jordan M. Stoyanov, zbMATH 1434.60003, 2020)

• Chennai Mathematical Institute, Siruseri, India

  Rajeeva L. Karandikar, B. V. Rao

Rajeeva Laxman Karandikar has been professor and director of Chennai Mathematical Institute, Tamil Nadu, India, since 2010. An Indian mathematician, statistician and psephologist, Prof. Karandikar is a fellow of the Indian Academy of Sciences, Bengaluru, India, and the Indian National Science Academy, New Delhi, India. He received his MStat and PhD from the Indian Statistical Institute, Kolkata, India, in 1978 and 1981, respectively. He spent two years as a visiting professor at the University of North Carolina, Chapel Hill, USA, and worked with Prof. Gopinath Kallianpur. He returned to the Indian Statistical Institute, New Delhi, India, in 1984. In 2006, he moved to Cranes Software International Limited, where he was executive vice president for analytics until 2010. His research interests include stochastic calculus, filtering theory, option pricing theory, psephology in the context of Indian elections and cryptography, among others.


B.V. Rao is an adjunct professor at Chennai Mathematical Institute, Tamil Nadu, India. He received his MSc degree in Statistics from Osmania University, Hyderabad, India, in 1965 and the doctoral degree from the Indian Statistical Institute, Kolkata, India, in 1970. His research interests include descriptive set theory, analysis, probability theory and stochastic calculus. He was a professor and later a distinguished scientist at the Indian Statistical Institute, Kolkata. Generations of Indian probabilists have benefitted from his teaching, where he taught from 1973 till 2009.

Policies and ethics